This lab is adapted from a Project Physics lab in the early 1980's. I am not sure if the acetates and photographs are still available.

The purpose of this lab will be to use the data from the Mars photographs to verify Kepler's three laws.

Directions

1. Obtain a large piece of graph paper from Mrs. Colwell. Carefully, determining the center of the paper. Place a large pencil dot in its position.

2. On this graph paper, draw a 10 cm radius circle as close to the center of the grid as possible using your compass. Label the center of this circle with a large "S" for the Sun. This circle represents the orbit of the Earth around the Sun.

3. From "S" draw a line to the right until it intersects with the circle. This represents the positive x-axis. Neatly label this intersection point as 0º - Sept. 23. Now draw a line from "S" to the left. This will represent the negative x-axis. Label this intersection point as 180º - March 21.

4. Since the Earth travels once (360º) around the Sun in 365 days, use the rough estimate that the Earth moves approximating 1 degree/day, to locate and neatly label each of the positions listed as A through P below on the circle drawn in Step #3.

Jan 4, 103.2º

April 6, 195.7º

July 5, 282.5º

Oct 4, 11.3º

Feb 4, 134.7º

May 6, 225º

Aug 5, 312.1º

Nov 3, 40º

March 7, 166º

June 5, 253.9º

Sept 4, 342º

Dec 4, 70.9º

Photograph

Date

Location on Earth's Orbit in Degrees

A

March 21

B

February 5

C

April 2O

D

March 8

E

May 26

F

April 12

G

September 16

H

August 4

I

November 22

J

October 11

K

January 21

L

December 9

M

March 19

N

February 3

O

April 4

P

February 21

5. After obtaining a booklet of star photographs and transparencies, locate Mars on each picture and use a short ruler to interpolate the planet's position. Record your results in the chart provided below. After checking with your instructor, share them with the rest of the class by placing them in the appropriate blanks on the board. Notice that the dates on which each of the pictures was taken coincides with the dates already placed on your Earth Orbit.

Photograph

Date

Location of Mars' Longitude in Degrees

A

March 21, 1931

B

February 5, 1933

C

April 2O, 1933

D

March 8, 1935

E

May 26, 1935

F

April 12, 1937

G

September 16, 1939

H

August 4, 1941

I

November 22, 1941

J

October 11, 1943

K

January 21, 1944

L

December 9, 1945

M

March 19, 1946

N

February 3, 1948

O

April 4, 1948

P

February 21, 1950

6. Through each Earth position, lightly sketch in a new "Oº axis" parallel to your original x-axis which passed through the Sun. Then using a protractor, locate the "line of sight" for Mars. Each set of overlays (AB), (CD), etc. represents one Martian year and will allow you to triangulate one Martian position - you will have a total of "8 spikes". Place a neat circle around each intersection and label them appropriately as MAB, MCD, MEF, MGH, MIJ, MKL, MMN, and MOP.

7. Use a ruler to draw in a straight line between two adjacent Mars positions. Then use a compass to bisect the line and a ruler to draw in the perpendicular bisector. Perform this operation up to four times for more accurate results. Extend the bisectors as long as necessary to insure that they intersect.

According to a theorem from geometry, the perpendicular bisectors of any two chords of a circle will intersect in the center of the circle. You should now be able to draw a circle that represents Mars' orbit. Note that the center of this circle will not pass through the Sun.

Calculations and Conclusions

1. Use two difference sets of Mars' dates to calculate the number of Earth days in one Mars' orbital period. State which sets you analyzed and the number of Earth days for each. Be careful of leap years - any year divisible by 4, 2008 is a leap year! Finally, average your two results and convert the result into years (1 Earth year = 365 days)

Set

Combination

days

1

2

What was the average period in "Earth days"?

What was the average period in "Earth years"?

2. Draw in the axis that connects the "center" of Mars' orbit with the Sun. Extend this line all the way across Mars' orbit. Now take three measurements: (1) the distance from the Sun to the "center" of Mars' orbit, (2) the distance from the Sun to the most distant edge of the axis, and (3) the distance from the Sun to the closest edge of the axis.

c = "center" of Mars' orbit to the Sun = ____ cm

RP = Perihelion radius = ____ cm

RA = Aphelion radius = ____ cm

3. An astronomical unit (AU) is defined as the average distance from the Earth to the Sun.

1 AU = Earth's radius = ____ cm.

4. Convert the three measurements in question #2: c, RP, and RA, to astronomical units (AU).

c = ____ AU

RP = Perihelion radius = ____ AU

RA = Aphelion radius = ____ AU

Verification of Kepler's 1st Law.

5. Your plot of Mars is an ellipse with respect to the Sun. We are now going to calculate the degree of its eccentricity.

According to astronomers, the eccentricity of Mars' orbit equals 0.093 Calculate a percent error for your calculed value for its eccentricity.

Verification of Kepler's 3rd Law.

6. Now that we have verified Kepler's 1st Law, we will move on to verify his 3rd Law by using your data to calculate Kepler's constant, k = T2/Rav2. Use your value for the average period, T, from question #1 and your value for Mars' average distance from the sun, a, from question #5.

k = ____

According to astronomers, Kepler's constant should be 1 for all satellites of the Sun. Calculate a percent error to determine the overall accuracy of your plot.

Verification of Kepler's 2nd Law.

Use carbon paper to trace onto a piece of cardboard the sector of the orbit defined
by the arc MGH and MIJ by first marking the points MIJ,
Sun, and MGH as well as the arc connecting MGH and MIJ.
Then remove the carbon paper. By using a ruler, draw in the radii between MGH
and the Sun and MIJ and the Sun. Cut out this sector and obtain its mass
in grams. Place your names on your cardboard, label your sector and record its mass
clearly.

Now repeat the above process but with the sector of the orbit defined by MEF
and MOP by marking the points MEF, Sun, and MOP
as well as the arc connecting MEF and MOP. Then remove the
carbon paper. By using a ruler, draw in the radii between MEF and the
Sun and MOP and the Sun. Cut out this sector and obtain its mass in grams.
Place your names on your cardboard, label your sector and record its mass clearly.

7. Using a piece of string or tape, measure the arc length of each sector. Next carefully
fold the arc length in half and measure its average, central, radius. Record your
information in the chart provided below and on each cardboard sector.

Sector

Arc length
(cm)

Average radius
(cm)

Mass
(g)

MGH and MIJ

MEF and MOP

8. What is the relationship between the relative masses of these sectors and their
relative areas?

Kepler's 2nd Law states that the ratio of v1 / v2 = R2 / R1
between any two points 1 and 2 on the same orbital path. If the time period for each of the sectors is equal, this relationship
can be rewritten as arc1 / arc2 = R2 / R1.

Calculate the ratio of these arc lengths: arcGH-IJ/ arcEF-OP

Calculate the ratio of these radii: radiusEF-OP/ radiusGH-IJ

Calculate the percentage difference between these values.

Staple your cardboard sectors to the back of the last page of your lab report.